Mathematics Colloquium/Taussky-Todd Lecture
l-adic cohomology has its origins in the study of congruences in the ring of integers and specifically in the problem of counting solutions of system of polynomial equations modulo a prime number q. This is a complex theory, conjectured by A. Weil, constructed by A. Grothendieck and developed by P. Deligne, N. Katz, G. Laumon and others. The basic objects are l-adic sheaves over an algebraic variety over Fq; to these are associated "trace functions" on the set of Fq-points. For the affine line, the functions can also be considered as q-periodic functions over the integers; they can then "interact" with the basic functions from analytic number theory, like the characteristic function of the primes. In this talk I will highlight some classical problems from analytic number theory in which sophisticated trace function pop-up naturally and will explain how basic and not so basic methods from analytic number theory and l-adic cohomology allow to measure these interactions.